Group #41014334 — subgroups

Subgroups — #41014334

Order 27008.

292 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#27212900	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₂₂	C₄	1	1	yes	characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₄₂₂	—	2	2	no	—
8	C₄×Dic₄₂₂	—	2	2	no	—
9	C₂×C₄⋊Dic₂₁₁	C₄	1	1	yes	characteristic
10	C₂×Dic₂₁₁⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₁₁	C₂²	1	2	yes	—
12	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
13	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₁₁	—	2	2	no	—
16	C₂×Dic₄₂₂	—	2	2	no	—
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	C₈	1	4	yes	complemented
19	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
21	C₂×C₂₁₁⋊C₈	—	2	2	no	—
22	C₂×C₂₁₁⋊C₈	—	2	4	no	—
23	C₄⋊Dic₂₁₁	—	2	2	no	—
24	Dic₂₁₁⋊C₄	—	2	4	no	—
25	C₄×Dic₂₁₁	—	2	2	no	—
26	C₄×Dic₂₁₁	—	2	2	no	—
27	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₁₁	—	2	2	no	—
31	C₂×C₈×C₂₁₁	—	2	2	no	—
32	C₂×C₈×C₂₁₁	Q₈	1	4	yes	complemented
33	C₄²×C₂₁₁	D₄	1	2	yes	—
34	C₄²×C₂₁₁	D₄	1	2	yes	—
35	Dic₄₂₂	—	2	8	no	—
36	C₂×Dic₂₁₁	—	2	2	no	—
37	C₂×Dic₂₁₁	—	2	2	no	—
38	C₂×Dic₂₁₁	—	2	2	no	—
39	C₂×Dic₂₁₁	—	2	4	no	—
40	C₂×Dic₂₁₁	C₂×C₈	1	4	yes	—
41	C₂₁₁⋊C₈	—	4	2	no	—
42	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	—	2	2	no	—
44	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₂₁₁	M₁₆	1	2	yes	—
48	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₁₁	#41	1	2	yes	—
52	C₈×C₂₁₁	—	2	4	no	—
53	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₁₁	C₄×Q₈	1	2	yes	—
55	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₂₁₁	#131	1	2	yes	—
58	C₄×C₂₁₁	—	2	2	no	—
59	C₄×C₂₁₁	C₂²⋊C₈	1	4	yes	—
60	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₁₁	#124	1	1	yes	characteristic
62	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₂₁₁	—	2	4	no	—
64	Dic₂₁₁	—	4	2	no	—
65	C₂×C₂₁₁	#391	1	1	yes	characteristic
66	C₂×C₂₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₂₁₁	#408	1	1	yes	characteristic
68	C₂×C₂₁₁	C₈×Q₈	1	2	yes	—
69	C₂×C₂₁₁	#446	1	2	yes	—
70	C₂₁₁	#1984	1	1	yes	characteristic, complemented, Sylow
71	#1984	—	211	1	no	maximal, Sylow
72	C₂×C₄×Q₈	—	211	1	no	—
73	C₂×C₄⋊C₈	—	211	1	no	—
74	C₂×C₄×C₈	D₂₁₁	1	1	yes	characteristic
75	#336	—	211	4	no	—
76	C₂²×Q₈	—	211	1	no	—
77	C₂²×C₈	—	211	2	no	—
78	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
79	C₄×Q₈	—	422	2	no	—
80	C₄×Q₈	—	422	2	no	—
81	C₂×C₄⋊C₄	—	211	2	no	—
82	C₂×C₄⋊C₄	—	211	1	no	—
83	C₂×C₄²	—	211	2	no	—
84	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
85	C₄⋊C₈	—	422	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₂×Q₈	—	422	2	no	—
88	C₂×Q₈	—	422	2	no	—
89	C₂×Q₈	—	211	4	no	—
90	C₂²×C₄	—	211	2	no	—
91	C₂²×C₄	—	211	2	no	—
92	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	422	2	no	—
98	C₂×C₈	—	422	4	no	—
99	C₂×C₈	Dic₄₂₂	1	4	yes	complemented
100	C₄⋊C₄	—	422	4	no	—
101	C₄⋊C₄	—	422	2	no	—
102	C₄²	—	422	2	no	—
103	C₄²	—	422	2	no	—
104	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
105	C₄²	D₈₄₄	1	2	yes	—
106	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
107	Q₈	—	422	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	422	2	no	—
110	C₂×C₄	—	422	4	no	—
111	C₂×C₄	—	422	2	no	—
112	C₂×C₄	—	422	2	no	—
113	C₂×C₄	—	211	4	no	—
114	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
115	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
116	C₂×C₄	C₈×D₂₁₁	1	2	yes	—
117	C₂×C₄	#16809945	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
121	C₂×C₄	#16809962	1	2	yes	—
122	C₈	—	2	4	no	—
123	C₈	—	844	2	no	—
124	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
125	C₂²	C₄×Dic₄₂₂	1	2	yes	—
126	C₂²	C₂×C₈×D₂₁₁	1	1	yes	characteristic
127	C₂²	C₂×#16809945	1	1	yes	characteristic
128	C₂²	#20269983	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	844	2	no	—
131	C₄	—	422	4	no	—
132	C₄	(C₂×D₂₁₁)⋊C₈	1	4	yes	—
133	C₄	#20269956	1	2	yes	—
134	C₄	#20269965	1	1	yes	characteristic
135	C₄	#20269977	1	1	yes	characteristic
136	C₂	#27213052	1	1	yes	characteristic
137	C₂	C₂×(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic
138	C₂	#27213112	1	1	yes	characteristic
139	C₂	C₈×Dic₄₂₂	1	2	yes	—
140	C₂	#27213276	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₄₂₂

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, characteristic

#27212900

C₂

yes

maximal

C₂×C₄×C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₄

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₄×Dic₄₂₂

—

C₄×Dic₄₂₂

—

C₂×C₄⋊Dic₂₁₁

C₄

yes

characteristic

C₂×Dic₂₁₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₈

yes

complemented

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

Dic₂₁₁⋊C₄

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

Q₈

yes

complemented

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₂×C₈

yes

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×C₈

yes

—

C₂×C₄×C₂₁₁

M₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×Q₈

yes

—

C₂²×C₂₁₁

C₂²×C₈

yes

characteristic

C₂²×C₂₁₁

C₂×M₁₆

yes

characteristic

C₂²×C₂₁₁

#131

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊C₈

yes

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#124

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#391

yes

characteristic

C₂×C₂₁₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₁₁

#408

yes

characteristic

C₂×C₂₁₁

C₈×Q₈

yes

—

C₂×C₂₁₁

#446

yes

—

C₂₁₁

#1984

yes

characteristic, complemented, Sylow

#1984

—

211

maximal, Sylow

C₂×C₄×Q₈

—

211

—

C₂×C₄⋊C₈

—

211

—

C₂×C₄×C₈

D₂₁₁

yes

characteristic

#336

—

211

—

C₂²×Q₈

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₄×Q₈

—

422

—

C₄×Q₈

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

C₄×C₈

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

Dic₄₂₂

yes

complemented

100

C₄⋊C₄

—

422

—

101

C₄⋊C₄

—

422

—

102

C₄²

—

422

—

103

C₄²

—

422

—

104

C₄²

C₂²⋊D₂₁₁

yes

—

105

C₄²

D₈₄₄

yes

—

106

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

107

Q₈

—

422

—

108

C₂×C₄

—

109

C₂×C₄

—

422

—

110

C₂×C₄

—

422

—

111

C₂×C₄

—

422

—

112

C₂×C₄

—

422

—

113

C₂×C₄

—

211

—

114

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

115

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

116

C₂×C₄

C₈×D₂₁₁

yes

—

117

C₂×C₄

#16809945

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

119

C₂×C₄

C₂×D₈₄₄

yes

characteristic

120

C₂×C₄

C₂×Dic₄₂₂

yes

—

121

C₂×C₄

#16809962

yes

—

122

C₈

—

123

C₈

—

844

—

124

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

125

C₂²

C₄×Dic₄₂₂

yes

—

126

C₂²

C₂×C₈×D₂₁₁

yes

characteristic

127

C₂²

C₂×#16809945

yes

characteristic

128

C₂²

#20269983

yes

—

129

C₄

—

130

C₄

—

844

—

131

C₄

—

422

—

132

C₄

(C₂×D₂₁₁)⋊C₈

yes

—

133

C₄

#20269956

yes

—

134

C₄

#20269965

yes

characteristic

135

C₄

#20269977

yes

characteristic

136

C₂

#27213052

yes

characteristic

137

C₂

C₂×(C₂×D₂₁₁)⋊C₈

yes

characteristic

138

C₂

#27213112

yes

characteristic

139

C₂

C₈×Dic₄₂₂

yes

—

140

C₂

#27213276

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂²×C₈
Fitting	C₂×C₄×C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#1984
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator

C₂²×C₂₁₁

C₂²×C₈

Fitting

C₂×C₄×C₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄×C₈

D₂₁₁

C₂₁₁

G ▷ C₂²×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
18. C₂×Dic₄₂₂	C₈	semidirect
32. C₂×C₈×C₂₁₁	Q₈	semidirect
70. C₂₁₁	#1984	semidirect
99. C₂×C₈	Dic₄₂₂	semidirect

H ≅ G/N

type

18. C₂×Dic₄₂₂

C₈

semidirect

32. C₂×C₈×C₂₁₁

Q₈

semidirect

70. C₂₁₁

#1984

semidirect

99. C₂×C₈

Dic₄₂₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×Dic₄₂₂	2
C₂×C₄⋊(C₂₁₁⋊C₈)	2
#27212900	2
C₂×C₄×C₈×C₂₁₁	2
#1984	211

subgroup

index

C₂×C₄×Dic₄₂₂

C₂×C₄⋊(C₂₁₁⋊C₈)

#27212900

C₂×C₄×C₈×C₂₁₁

#1984

211

Subgroups — #41014334

Conjugacy classes of subgroups

Complements

Maximal subgroups